Rocky Mountain Journal of Mathematics Articles (Project Euclid)
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The latest articles from Rocky Mountain Journal of Mathematics on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTMon, 02 May 2011 10:11 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Rings Over which All Modules are Strongly Gorenstein Projective
http://projecteuclid.org/euclid.rmjm/1277385512
<strong>Driss Bennis</strong>, <strong>Najib Mahdoua</strong>, <strong>Khalid Ouarghi</strong><p><strong>Source: </strong>Rocky Mountain J. Math., Volume 40, Number 3, 749--759.</p>projecteuclid.org/euclid.rmjm/1277385512_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTInfluence of mean curvature on Mountain-pass solutions for Hardy-Sobolev equationshttps://projecteuclid.org/euclid.rmjm/1561318390<strong>Hassan Jaber</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 2, 505--519.</p><p><strong>Abstract:</strong><br/>
Let $\Omega $ be a smooth open domain in $\mathbb{R} ^n$, $n \geq 3$, with $0 \in \partial \Omega $ and $s \in {]}0,2{[}$. Let $2^\star(s) := {2(n-s)}/({n-2})$ be the critical Hardy-Sobolev exponent. Consider the Dirichlet problem that corresponds to the critical and perturbative Hardy-Sobolev equation \begin{aligned}\begin{cases}-\Delta u = {u^{2^\star(s) -1}}/{|x|^s} + h u^{q-1} \quad &\mbox {in } \Omega ,\\u \equiv 0 &\mbox {on } \partial \Omega, \end{cases}\end{aligned} where $q \in {]} 2,2^\star {[}$ with $2^\star := 2^\star (0)$ and $h \in C^0(\overline {\Omega })$, $h \geq 0$, almost everywhere on $\Omega $. In this article, we investigate the influence of perturbation and mean curvature at a boundary singularity on the existence of a positive Mountain pass solution to the above equation.
</p>projecteuclid.org/euclid.rmjm/1561318390_20190623153319Sun, 23 Jun 2019 15:33 EDTSome identities involving special numbers and moments of random variableshttps://projecteuclid.org/euclid.rmjm/1561318391<strong>Taekyun Kim</strong>, <strong>Yonghong Yao</strong>, <strong>Dae San Kim</strong>, <strong>Hyuck-In Kwon</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 2, 521--538.</p><p><strong>Abstract:</strong><br/>
In this paper, we derive some identities involving special numbers and moments of random variables by using the generating functions of the moments of certain random variables. Here, the related special numbers are Stirling numbers of the first and second kinds, degenerate Stirling numbers of the first and second kinds, derangement numbers, higher-order Bernoulli numbers and Bernoulli numbers of the second kind.
</p>projecteuclid.org/euclid.rmjm/1561318391_20190623153319Sun, 23 Jun 2019 15:33 EDTThe Besicovitch covering lemma and maximal functionshttps://projecteuclid.org/euclid.rmjm/1561318392<strong>Steven G. Krantz</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 2, 539--555.</p><p><strong>Abstract:</strong><br/>
This paper has two purposes. First, we explain and describe the Besicovitch covering lemma, and we provide a new proof. Applications are given, particularly to the ideas of Nagel and Stein about Fatou theorems through approach regions which are not nontangential. Second, we examine the strong maximal function and give a new, simple, geometric proof of its $L^p$ boundedness.
</p>projecteuclid.org/euclid.rmjm/1561318392_20190623153319Sun, 23 Jun 2019 15:33 EDTOn global $\mathscr C$-dimensionshttps://projecteuclid.org/euclid.rmjm/1561318393<strong>Weiqing Li</strong>, <strong>Liang Yan</strong>, <strong>Baiyu Ouyang</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 2, 557--577.</p><p><strong>Abstract:</strong><br/>
Let $R$ be a ring and $\mathcal {M}$ the left or right $R$-module category. Let $\mathscr {C}$ be a class of $R$-modules, closed under extensions, finite direct sums, direct summands and isomorphisms. Suppose that $\mathscr {C}$ is both precovering and preenveloping. We prove that every $m$th $\mathscr {C}$-cosyzygy of any $R$-module is contained in $\mathscr {C}$ if and only if every $m$th $\mathscr {C}$-syzygy of any $R$-module is contained in $\mathscr {C}$; if $\mathscr {C}$ is closed under cokernels of monomorphisms, then gl right $\mathscr {C}$-${\dim }\, {\mathcal {M}}\leq m$ and every $m$th and $(m+1)$th $\mathscr {C}$-cosyzygy of any $R$-module have a monic $\mathscr {C}$-preenvelope if and only if gl left $\mathscr {C}$-${\dim }\, {\mathcal {M}} \leq m-2$, $m\geq 2$; if $\mathscr {C}$ is closed under kernels of epimorphisms, then gl left $\mathscr {C}$-${\dim }\, {\mathcal {M}}\leq m$ and every $m$th and $(m+1)$th $\mathscr {C}$-syzygy of any $R$-module have an epic $\mathscr {C}$-precover if and only if gl right $\mathscr {C}$-${\dim }\,{\mathcal {M}}\leq m-2$, $m\geq 2$; if every nonzero $R$-module has a nonzero $\mathscr {C}$-preenvelope and a nonzero $\mathscr {C}$-precover, then gl right $\mathscr {C}$-${\dim }\, {\mathcal {M}} =$ gl left $\mathscr {C}$-${\dim }\, {\mathcal {M}}$. Some applications are given. Some known results are extended or improved.
</p>projecteuclid.org/euclid.rmjm/1561318393_20190623153319Sun, 23 Jun 2019 15:33 EDTIntegrability of reversible and equivariant quadratic polynomial differential systems in the planehttps://projecteuclid.org/euclid.rmjm/1561318394<strong>Jaume Llibre</strong>, <strong>Claudia Valls</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 2, 579--591.</p><p><strong>Abstract:</strong><br/>
We study the existence of first integrals for the class of reversible and equivariant quadratic polynomial differential systems in the plane. We put special emphasis in the study of the analytic first integrals.
</p>projecteuclid.org/euclid.rmjm/1561318394_20190623153319Sun, 23 Jun 2019 15:33 EDTBi-additive $s$-functional inequalities and quasi-multipliers on Banach algebrashttps://projecteuclid.org/euclid.rmjm/1561318395<strong>Choonkil Park</strong>, <strong>Yuanfeng Jin</strong>, <strong>Xiaohong Zhang</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 2, 593--607.</p><p><strong>Abstract:</strong><br/>
In this paper, we solve the following bi-additive $s$-functional inequalities: $$ \displaylines { \, \| f(x+y, z-w) + f(x-y, z+w) -2f(x,z)+2 f(y, w)\| \hfill \cr \hfill \quad \le \Bigl \|s \Bigl (2f\Bigl (\frac {x+y}{2}, z-w\Bigr ) + 2f\Bigl (\frac {x-y}{2}, z+w\Bigr ) - 2f(x,z )+ 2 f(y, w)\Bigr )\Bigr \| ,\! \cr \, \Bigl \|2f\Bigl (\frac {x+y}{2}, z-w\Bigr ) +2 f\Bigl (\frac {x-y}{2}, z+w\Bigr ) -2 f(x,z )+2 f(y, w)\Bigr \| \hfill \cr \hfill \le \|s ( f(x+y, z-w) + f(x-y, z+w) -2f(x,z) +2 f(y, w) )\|, \cr \, } $$ where $s$ is a fixed nonzero complex number with $|s |\lt 1$. We also prove the Hyers-Ulam stability of quasi-multipliers on Banach algebras and unital $C^*$-algebras associated with the bi-additive $s$-functional inequalities above.
</p>projecteuclid.org/euclid.rmjm/1561318395_20190623153319Sun, 23 Jun 2019 15:33 EDTMultiple positive solutions for a coupled system of $p$-Laplacian fractional order three-point boundary value problemshttps://projecteuclid.org/euclid.rmjm/1561318396<strong>S. Nageswara Rao</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 2, 609--626.</p><p><strong>Abstract:</strong><br/>
In this paper, we attempt to establish the existence of at least three positive solutions for a coupled system of $p$-Laplacian fractional order boundary value problems \begin{aligned}\begin{cases} D_{a^+}^{\beta _1}(\phi _{p}(D_{a^+}^{\alpha _1}u(t)))=f_1(t, u(t), v(t)) & \quad a\lt t\lt b,\\ D_{a^+}^{\beta _2}(\phi _{p}(D_{a^+}^{\alpha _2}v(t)))=f_2(t, u(t), v(t)) & \quad a\lt t\lt b, \end{cases}\end{aligned} with the boundary conditions \begin{aligned}\begin{cases} u^{(j)}(a)=0,\ j=0,1,2,\ldots , \ u{''}(b)=\delta u{''}(\xi ),\\ \phi _{p}(D_{a^+}^{\alpha _1}u(a))=0, \ \phi _{p}(D_{a^+}^{\alpha _1} u(b)) =\vartheta \phi _{p}( D_{a^+}^{\alpha _1}u(\eta )),\\ v^{(j)}(a)=0,\ j=0,1,2,\ldots , \ v{''}(b)=\delta v{''}(\xi ),\\ \phi _{p}(D_{a^+}^{\alpha _2}v(a))=0, \ \phi _{p}(D_{a^+}^{\alpha _2} v(b))=\vartheta \phi _{p}( D_{a^+}^{\alpha _2}v(\eta )), \end{cases}\end{aligned} by applying the five functional fixed point theorem.
</p>projecteuclid.org/euclid.rmjm/1561318396_20190623153319Sun, 23 Jun 2019 15:33 EDTDerivatives of Blaschke products in weighted mixed norm spaceshttps://projecteuclid.org/euclid.rmjm/1561318397<strong>Atte Reijonen</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 2, 627--643.</p><p><strong>Abstract:</strong><br/>
For $1/2\lt p\lt \infty $, $0\lt q\lt \infty $ and a certain two-sided doubling weight $\omega $, we give a condition for the zeros of a Blaschke product $B$ which guarantees that $$ \|B'\|_{A^{p,q}_\omega }^q=\int _0^1 \bigg (\int _0^{2\pi } |B'(re^{i\theta })|^p d\theta \bigg )^{q/p} \omega (r)\,dr\lt \infty . $$ In addition, it is shown that the condition is necessary if the zero-sequence is a finite union of separated sequences.
</p>projecteuclid.org/euclid.rmjm/1561318397_20190623153319Sun, 23 Jun 2019 15:33 EDTPartial group algebra with projections and relationshttps://projecteuclid.org/euclid.rmjm/1561318398<strong>Danilo Royer</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 2, 645--660.</p><p><strong>Abstract:</strong><br/>
We introduce the notion of the partial group algebra with projections and relations and show that this $C^*$-algebra is a partial crossed product. Examples of partial group algebras with projections and relations are the Cuntz-Krieger algebras and the unitization of $C^*$-algebras of directed graphs.
</p>projecteuclid.org/euclid.rmjm/1561318398_20190623153319Sun, 23 Jun 2019 15:33 EDTAsymptotics for Hawkes processes with large and small baseline intensitieshttps://projecteuclid.org/euclid.rmjm/1561318399<strong>Youngsoo Seol</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 2, 661--680.</p><p><strong>Abstract:</strong><br/>
This paper focuses on asymptotic results for linear Hawkes processes with large and small baseline intensities. The intensity process is one of the main tools used to work with the dynamical properties of a general point process. It is of essential interest in credit risk study, in particular. First, we establish a large deviation principle and a moderate deviation principle for the Hawkes process with large baseline intensity. In addition, a law of large numbers and a central limit theorem are also obtained. Second, we observe asymptotic behaviors for the Hawkes process with small baseline intensity. The main idea of the proof relies on the immigration-birth representation and the observations of the moment generating function for the linear Hawkes process.
</p>projecteuclid.org/euclid.rmjm/1561318399_20190623153319Sun, 23 Jun 2019 15:33 EDTSome new generalizations and applications of the Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomialshttps://projecteuclid.org/euclid.rmjm/1561318400<strong>H.M. Srivastava</strong>, <strong>M. Masjed-Jamei</strong>, <strong>M.R. Beyki</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 2, 681--697.</p><p><strong>Abstract:</strong><br/>
By means of six specific generating functions, we introduce a type of generalized parametric Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials and systematically study their basic properties. As an application of the new polynomials, we use them in computing a new series of Taylor-type.
</p>projecteuclid.org/euclid.rmjm/1561318400_20190623153319Sun, 23 Jun 2019 15:33 EDTRegularity of powers of edge ideals of unicyclic graphshttps://projecteuclid.org/euclid.rmjm/1563847229<strong>Ali Alilooee</strong>, <strong>Selvi Kara Beyarslan</strong>, <strong>S. Selvaraja</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 3, 699--728.</p><p><strong>Abstract:</strong><br/>
Let $G$ be a finite simple graph and $I(G)$ denote the corresponding edge ideal. In this paper, we prove that, if $G$ is a unicyclic graph, then, for all $s \geq 1$, the regularity of $I(G)^s$ is exactly $2s+\DeclareMathOperator{reg} (I(G))-2$. We also give a combinatorial characterization of unicyclic graphs with regularity $\nu (G)+1$ and $\nu (G)+2$, where $\nu (G)$ denotes the induced matching number of $G$.
</p>projecteuclid.org/euclid.rmjm/1563847229_20190722220120Mon, 22 Jul 2019 22:01 EDTSets of lengths of powers of a variablehttps://projecteuclid.org/euclid.rmjm/1563847230<strong>Richard Belshoff</strong>, <strong>Daniel Kline</strong>, <strong>Mark W. Rogers</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 3, 729--741.</p><p><strong>Abstract:</strong><br/>
A positive integer $k$ is a length of a polynomial if that polynomial factors into a product of $k$ irreducible polynomials. We find the set of lengths of polynomials of the form $x^n$ in $R[x]$, where $(R, \mathfrak{m} )$ is an Artinian local ring with $\mathfrak{m} ^2=0$.
</p>projecteuclid.org/euclid.rmjm/1563847230_20190722220120Mon, 22 Jul 2019 22:01 EDTOn fractionally dense setshttps://projecteuclid.org/euclid.rmjm/1563847231<strong>Jaitra Chattopadhyay</strong>, <strong>Bidisha Roy</strong>, <strong>Subha Sarkar</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 3, 743--760.</p><p><strong>Abstract:</strong><br/>
In this article, we prove that some subsets of the set of natural numbers $\mathbb {N}$ and any non-zero ideals of an order of an imaginary quadratic field are fractionally dense in $\mathbb {R}_{>0}$ and $\mathbb {C}$, respectively.
</p>projecteuclid.org/euclid.rmjm/1563847231_20190722220120Mon, 22 Jul 2019 22:01 EDTCharacterization of Lie multiplicative derivation on alternative ringshttps://projecteuclid.org/euclid.rmjm/1563847232<strong>Bruno Leonardo Macedo Ferreira</strong>, <strong>Henrique Guzzo Jr</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 3, 761--772.</p><p><strong>Abstract:</strong><br/>
In this paper, we generalize the result valid for associative rings Bresar and Martindale III to alternative rings. Let $\mathfrak{R} $ be a unital alternative ring, and $\mathfrak{D} \colon \mathfrak{R} \rightarrow \mathfrak{R} $ is a Lie multiplicative derivation. Then, $\mathfrak{D} $ is the form $\delta + \tau $, where $\delta $ is an additive derivation of $\mathfrak{R} $ and $\tau $ is a map from $\mathfrak{R} $ into its center $\mathcal {\mathfrak{R} }$, which maps commutators into the zero.
</p>projecteuclid.org/euclid.rmjm/1563847232_20190722220120Mon, 22 Jul 2019 22:01 EDTLagrange's theorem for Hom-Groupshttps://projecteuclid.org/euclid.rmjm/1563847233<strong>Mohammad Hassanzadeh</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 3, 773--787.</p><p><strong>Abstract:</strong><br/>
Hom-groups are nonassociative generalizations of groups where the unitality and associativity are twisted by a map. We show that a Hom-group $(G, \alpha )$ is a pointed idempotent quasigroup (pique). We use Cayley tables of quasigroups to introduce some examples of Hom-groups. Introducing the notions of Hom-subgroups and cosets we prove Lagrange's theorem for finite Hom-groups. This states that the order of any Hom-subgroup $H$ of a finite Hom-group $G$ divides the order of $G$. We linearize Hom-groups to obtain a class of nonassociative Hopf algebras called Hom-Hopf algebras. As an application of our results, we show that the dimension of a Hom-sub-Hopf algebra of the finite dimensional Hom-group Hopf algebra $\mathbb {K}G$ divides the order of $G$. The new tools introduced in this paper could potentially have applications in theories of quasigroups, nonassociative Hopf algebras, Hom-type objects, combinatorics, and cryptography.
</p>projecteuclid.org/euclid.rmjm/1563847233_20190722220120Mon, 22 Jul 2019 22:01 EDTSymmetry and nonexistence results for a fractional Hénon-Hardy system on a half-spacehttps://projecteuclid.org/euclid.rmjm/1563847234<strong>Anh Tuan Duong</strong>, <strong>Phuong Le</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 3, 789--816.</p><p><strong>Abstract:</strong><br/>
We study the fractional Henon-Hardy system \begin{aligned}\begin{cases}(-\Delta )^{s/2} u(x) = |x|^\alpha v^p(x), & x\in \mathbb{R}^n_+, \\(-\Delta )^{s/2} v(x) = |x|^\beta u^q(x), & x\in \mathbb{R}^n_+, \\ u(x)=v(x)=0, & x\in \mathbb{R}^n\setminus \mathbb{R}^n_+,\end{cases}\end{aligned} where $n\ge 2$, $0\lt s\lt 2$, $\alpha ,\beta >-s$ and $p,q\ge 1$. We also consider an equivalent integral system. By using a direct method of moving planes, we prove some symmetry and nonexistence results for positive solutions under various assumptions on $\alpha $, $\beta $, $p$ and $q$.
</p>projecteuclid.org/euclid.rmjm/1563847234_20190722220120Mon, 22 Jul 2019 22:01 EDTThermal avalanche after non-simultaneous blow-up in heat equations coupled via nonlinear boundaryhttps://projecteuclid.org/euclid.rmjm/1563847235<strong>Bingchen Liu</strong>, <strong>Fengjie Li</strong>, <strong>Mengzhen Dong</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 3, 817--847.</p><p><strong>Abstract:</strong><br/>
In this paper, we study a parabolic problem defined in the half line coupled via exponential-type boundary flux. Firstly, we prove the optimal classification of the two components of blow-up solutions when time reaches the blow-up time from left. Blow-up takes place only at the origin, and simultaneous blow-up rates are determined as well. Secondly, we study the weak extension after the blow-up time. Complete blow-up always occurs whether simultaneous blow-up arises or not. Moreover, an instantaneous propagation of the blow-up singularity to the whole spatial domain occurs at the blow-up time, which is the so-called thermal avalanche phenomenon. Finally, we use the evolution of the $k$-level set of solutions in the approximations to characterize the propagation of the singularity.
</p>projecteuclid.org/euclid.rmjm/1563847235_20190722220120Mon, 22 Jul 2019 22:01 EDTMaximal chains of prime ideals of different lengths in unique factorization domainshttps://projecteuclid.org/euclid.rmjm/1563847236<strong>Susan Loepp</strong>, <strong>Alex Semendinger</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 3, 849--865.</p><p><strong>Abstract:</strong><br/>
We show that, given integers $n_1,n_2, \ldots ,n_k$ with $2 \lt n_1 \lt n_2 \lt \cdots \lt n_k$, there exists a local (Noetherian) unique factorization domain that has maximal chains of prime ideals of lengths $n_1, n_2, \ldots ,n_k$ which are disjoint except at their minimal and maximal elements. In addition, we demonstrate that unique factorization domains can have other unusual prime ideal structures.
</p>projecteuclid.org/euclid.rmjm/1563847236_20190722220120Mon, 22 Jul 2019 22:01 EDTInterpolation for second order stationary random fields: time domain recipehttps://projecteuclid.org/euclid.rmjm/1563847237<strong>Z. Mafakheri</strong>, <strong>A.R. Soltani</strong>, <strong>Z. Shishebor</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 3, 867--885.</p><p><strong>Abstract:</strong><br/>
We consider a discrete time second order stationary random field and provide a time domain recipe for the interpolation based on the southwest and northeast corners. Our method is based on Salehi's approach, applying Von Neumann's celebrated alternative projection for-\break \noindent mula, but making a short cut by interpolating the innovations in the forward and backward moving average representations. We provide explicit expressions for the interpolator and error terms for the moving average random fields of finite order; for the MA($\boldsymbol {1}$) spatial model, we express the interpolator in terms of the observed values and the coefficients of the model. Following Kohli and Pourahmadi, we also derive the covariances between the present values and interpolation errors.
</p>projecteuclid.org/euclid.rmjm/1563847237_20190722220120Mon, 22 Jul 2019 22:01 EDTAn upper bound for the moments of a GCD related to Lucas sequenceshttps://projecteuclid.org/euclid.rmjm/1563847238<strong>Daniele Mastrostefano</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 3, 887--902.</p><p><strong>Abstract:</strong><br/>
Let $(u_n)_{n \geq 0}$ be a non-degenerate Lucas sequence, given by the relation $u_n=a_1 u_{n-1}+a_2 u_{n-2}$. Let $\ell _u(m)={\rm lcm}(m, z_u(m))$, for $(m,a_2)=1$, where $z_u(m)$ is the rank of appearance of $m$ in $u_n$. We prove that $$ \sum _{\substack {m>x\\ (m,a_2)=1}}\frac {1}{\ell _u(m)}\leq \exp \biggl (\!-\biggl (\frac 1{\sqrt {6}}-\varepsilon +o(1)\biggr )\sqrt {(\log x)(\log \log x)}\biggr ), $$ when $x$ is sufficiently large in terms of $\varepsilon $, and where the $o(1)$ depends on $u$. Moreover, if $g_u(n)=\gcd (n,u_n)$, we show that for every $k\geq 1$, $$ \sum _{n\leq x}g_u(n)^{k}\leq x^{k+1}\exp (-(1+o(1)) \sqrt {(\log x)(\log \log x)}), $$ when $x$ is sufficiently large, and where the $o(1)$ depends upon $u$ and $k$. This gives a partial answer to a question posed by C. Sanna. As a by-product, we derive bounds on $\#\{n\leq x: (n, u_n)>y\}$, at least in certain ranges of $y$, which strengthens what was already obtained by Sanna. Finally, we begin the study of the multiplicative analogs of $\ell _u(m)$, finding interesting results.
</p>projecteuclid.org/euclid.rmjm/1563847238_20190722220120Mon, 22 Jul 2019 22:01 EDTMore measure and category questions for subsequences of a given sequencehttps://projecteuclid.org/euclid.rmjm/1563847239<strong>Harry I. Miller</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 3, 903--911.</p><p><strong>Abstract:</strong><br/>
The author for several years has considered two problems: the first involving various gauges of the size of sets of real numbers, and the second, with properties of the class of all subsequences of a given sequence. This work continues the earlier papers connecting these two areas.
</p>projecteuclid.org/euclid.rmjm/1563847239_20190722220120Mon, 22 Jul 2019 22:01 EDTIncomplete hypergeometric systems associated to $1$-simplex $\times $ $(n-1)$-simplexhttps://projecteuclid.org/euclid.rmjm/1563847240<strong>Kenta Nishiyama</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 3, 913--928.</p><p><strong>Abstract:</strong><br/>
The ${\cal A}$-hypergeometric system was introduced by Gel'fand, Kapranov and Zelevinsky in the 1980's. Among several classes of ${\cal A}$-hypergeometric functions, those for $1$-simplex $\times $ $(n{-}1)$-simplex are known to be a very nice class. We will study an incomplete analog of this class.
</p>projecteuclid.org/euclid.rmjm/1563847240_20190722220120Mon, 22 Jul 2019 22:01 EDTInvertibility of operators on atomic subspaces of $L^1$ and an application to the Neumann problemhttps://projecteuclid.org/euclid.rmjm/1563847241<strong>Hugo Ocampo-Salgado</strong>, <strong>Jorge Rivera-Noriega</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 3, 929--944.</p><p><strong>Abstract:</strong><br/>
We prove a criterion for invertibility of operators on adequate adaptations to the boundary of a smooth domain of atomic subspaces of $L^1$, originally defined on ${\mathbb{R}^n} $ by Sweezy. As an application, we establish solvability of the Neumann problem for harmonic functions on smooth domains, assuming that the normal derivative belongs to said atomic subspaces of $L^1$.
</p>projecteuclid.org/euclid.rmjm/1563847241_20190722220120Mon, 22 Jul 2019 22:01 EDTThe Kusuoka measure and the energy Laplacian on level-$k$ Sierpiński gasketshttps://projecteuclid.org/euclid.rmjm/1563847242<strong>Anders Öberg</strong>, <strong>Konstantinos Tsougkas</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 3, 945--961.</p><p><strong>Abstract:</strong><br/>
We extend and survey results in the theory of analysis on fractal sets from the standard Laplacian on the Sierpinski gasket to the energy Laplacian, which is defined weakly by using the Kusuoka energy measure. We also extend results from the Sierpinski gasket to level-$k$ Sierpinski gaskets, for all $k\geq 2$. We observe that the pointwise formula for the energy Laplacian is valid for all level-$k$ Sierpinski gaskets, $SG_k$, and we provide a proof of a known formula for the renormalization constants of the Dirichlet form for post-critically finite self-similar sets along with a probabilistic interpretation of the Laplacian pointwise formula. We also provide a vector self-similar formula and a variable weight self-similar formula for the Kusuoka measure on $SG_k$, as well as a formula for the scaling of the energy Laplacian.
</p>projecteuclid.org/euclid.rmjm/1563847242_20190722220120Mon, 22 Jul 2019 22:01 EDTSeshadri constants and special configurations of points in the projective planehttps://projecteuclid.org/euclid.rmjm/1563847243<strong>Piotr Pokora</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 3, 963--978.</p><p><strong>Abstract:</strong><br/>
In the present note, we focus on certain properties of special curves that might be used in the theory of multi-point Seshadri constants for ample line bundles on the complex projective plane. In particular, we provide three Ein-Lazarsfeld-Xu-type lemmas for plane curves and a lower bound on the multi-point Seshadri constant of $\mathcal {O}_{\mathbb {P}^{2}}(1)$ under the assumption that the chosen points are not very general. In the second part, we focus on certain arrangements of points in the plane which are given by line arrangements. We show that, in some cases, the multi-point Seshadri constants of $\mathcal {O}_{\mathbb {P}^{2}}(1)$ centered at singular loci of line arrangements are computed by lines from the arrangement having some extremal properties.
</p>projecteuclid.org/euclid.rmjm/1563847243_20190722220120Mon, 22 Jul 2019 22:01 EDTWaring's Theorem revisitedhttps://projecteuclid.org/euclid.rmjm/1563847244<strong>Andrés Rojas</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 3, 979--1003.</p><p><strong>Abstract:</strong><br/>
This paper consists in a revision and extension of a classic result, Waring's Theorem, about the barycenter of the intersection points of two plane algebraic curves. The theorem arises from the study of the parts with highest degree of the equation of a curve, which are completely determined by the barycentric parallel lines of the groups of asymptotes.
</p>projecteuclid.org/euclid.rmjm/1563847244_20190722220120Mon, 22 Jul 2019 22:01 EDTLow regularity ray tracing for wave equations with Gaussian beamshttps://projecteuclid.org/euclid.rmjm/1563847245<strong>Alden Waters</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 3, 1005--1027.</p><p><strong>Abstract:</strong><br/>
We prove observability estimates for oscillatory Cauchy data modulo a small kernel for $n$-dimensional wave equations with space and time dependent $C^2$ and $C^{1,1}$ coefficients using Gaussian beams. We assume the domains and observability regions are in $\mathbb {R}^n$, and the GCC applies. This work generalizes previous observability estimates to higher dimensions and time dependent coefficients. The construction for the Gaussian beamlets solving $C^{1,1}$ wave equations represents an improvement and simplification over Waters (2011).
</p>projecteuclid.org/euclid.rmjm/1563847245_20190722220120Mon, 22 Jul 2019 22:01 EDTA geometric full multigrid method and fourth-order compact scheme for the 3D Helmholtz equation on nonuniform grid discretizationhttps://projecteuclid.org/euclid.rmjm/1563847246<strong>Zhenwei Yang</strong>, <strong>Xiaobin Li</strong>, <strong>Lei Feng</strong>, <strong>Xinxin Hou</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 3, 1029--1047.</p><p><strong>Abstract:</strong><br/>
A full multigrid method and fourth-order compact difference scheme is designed to solve the 3D Helmholtz equation on unequal mesh size. Three dimensional restriction and prolongation operators of the multigrid method on unequal grids could be constructed based on volume law. Two numerical experiments are implemented, and the results show the computational efficiency and accuracy of the full multigrid method. The study also illustrates that the full multigrid method with fourth-order compact difference scheme has great advantages in computation, which has been time consuming, and in iterative convergence efficiency.
</p>projecteuclid.org/euclid.rmjm/1563847246_20190722220120Mon, 22 Jul 2019 22:01 EDTIdentities for the zeros of entire functions of finite rank and spectral theoryhttps://projecteuclid.org/euclid.rmjm/1567044023<strong>N. Anghel</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 4, 1049--1062.</p><p><strong>Abstract:</strong><br/>
A theorem of Gil', relating the zeros of entire functions of finite order to traces of powers of matrices, is generalized to entire functions of finite rank and then analyzed from the point of view of spectral theory. Plenty of relevant examples are given, including a generalization of Viete's relations for the elementary symmetric functions of the roots of a polynomial.
</p>projecteuclid.org/euclid.rmjm/1567044023_20190828220100Wed, 28 Aug 2019 22:01 EDTFunctions analytic in the unit ball having bounded $L$-index in a directionhttps://projecteuclid.org/euclid.rmjm/1567044028<strong>Andriy Bandura</strong>, <strong>Oleh Skaskiv</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 4, 1063--1092.</p><p><strong>Abstract:</strong><br/>
We propose a generalization of a concept of bounded index for analytic functions in the unit ball. Use of directional derivatives gives us a possibility to deduce the necessary and sufficient conditions of boundedness of $L$-index in a direction for analytic functions of several variables, namely, we obtain an analog of Hayman's theorem and a logarithmic criteria for this class. The criteria describe the behavior of the directional logarithmic derivative outside the zero set and a uniform distribution of zeros in some sense. The criteria are useful for studying analytic solutions of partial differential equations and estimating their growth. We present a scheme of this application.
</p>projecteuclid.org/euclid.rmjm/1567044028_20190828220100Wed, 28 Aug 2019 22:01 EDTAn explicit conductor formula for $\operatorname{GL} _{n} \times \operatorname{GL} _{1}$https://projecteuclid.org/euclid.rmjm/1567044029<strong>Andrew Corbett</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 4, 1093--1110.</p><p><strong>Abstract:</strong><br/>
We prove an explicit formula for the conductor of an irreducible, admissible representation of $\operatorname{GL} _{n}(F)$ twisted by a character of $F^{\times} $ where the field $F$ is local and non-archimedean. As a consequence, we quantify the number of character twists of such a representation of fixed conductor.
</p>projecteuclid.org/euclid.rmjm/1567044029_20190828220100Wed, 28 Aug 2019 22:01 EDTUnlabeled signed graph coloringhttps://projecteuclid.org/euclid.rmjm/1567044030<strong>Brian Davis</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 4, 1111--1122.</p><p><strong>Abstract:</strong><br/>
We extend the work of Hanlon on the chromatic polynomial of an unlabeled graph to define the unlabeled chromatic polynomial of an unlabeled signed graph. Explicit formulas are presented for labeled and unlabeled signed chromatic polynomials as summations over distinguished order-ideals of the signed partition lattice. We also define the quotient of a signed graph by a signed permutation, and show that its signed graphic arrangement is closely related to an induced arrangement on a distinguished subspace. Lastly, a formula for the number of unlabeled acyclic orientations of a signed graph is presented which recalls classical reciprocity theorems of Stanley and Zaslavsky.
</p>projecteuclid.org/euclid.rmjm/1567044030_20190828220100Wed, 28 Aug 2019 22:01 EDTFinite subgroups of the extended modular grouphttps://projecteuclid.org/euclid.rmjm/1567044031<strong>Gregory Dresden</strong>, <strong>Prakriti Panthi</strong>, <strong>Anukriti Shrestha</strong>, <strong>Jiahao Zhang</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 4, 1123--1127.</p><p><strong>Abstract:</strong><br/>
We show that in the extended modular group $\overline \Gamma = \rm {PGL}(2, \mathbb {Z})$ there are exactly seven finite subgroups up to conjugacy; three subgroups of size 2, one subgroup each of size 3, 4, and 6, and the trivial subgroup of size 1.
</p>projecteuclid.org/euclid.rmjm/1567044031_20190828220100Wed, 28 Aug 2019 22:01 EDTEssentially hyponormal weighted composition operators on the Hardy and weighted Bergman spaceshttps://projecteuclid.org/euclid.rmjm/1567044032<strong>Mahsa Fatehi</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 4, 1129--1142.</p><p><strong>Abstract:</strong><br/>
Let $\varphi $ be an analytic self-map of the open unit disk $\mathbb {D}$ and let $\psi $ be an analytic function on $\mathbb {D}$ such that the weighted composition operator $C_{\psi ,\varphi }$ defined by $C_{\psi ,\varphi }(f)=\psi f\circ \varphi $ is bounded on the Hardy and weighted Bergman spaces. We characterize those weighted composition operators $C_{\psi ,\varphi }$ on $H^{2}$ and $A_{\alpha }^{2}$ that are essentially hypo-normal, when $\varphi $ is a linear-fractional non-automorphism.
</p>projecteuclid.org/euclid.rmjm/1567044032_20190828220100Wed, 28 Aug 2019 22:01 EDTOpen book embeddings of closed non-orientable $3$-manifoldshttps://projecteuclid.org/euclid.rmjm/1567044033<strong>Abhijeet Ghanwat</strong>, <strong>Suhas Pandit</strong>, <strong>Selvakumar A</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 4, 1143--1168.</p><p><strong>Abstract:</strong><br/>
In this note, we discuss open book embeddings of closed non-orientable $3$-manifolds in $5$-manifolds. We also show that a huge class of closed orientable $3$-manifolds, namely the orientation double covers of certain closed non-orientable $3$-manifolds, open book embeds in the $5$-sphere $S^5$. Finally, we give a new proof of a well-known theorem which states that every closed non-orientable $3$-manifold smoothly embeds in $S^5$.
</p>projecteuclid.org/euclid.rmjm/1567044033_20190828220100Wed, 28 Aug 2019 22:01 EDTMultiple positive solutions for a $(p,q)$-Laplace equation involving singular and critical termshttps://projecteuclid.org/euclid.rmjm/1567044034<strong>Liu-Tao Guo</strong>, <strong>Hong-Min Suo</strong>, <strong>Chang-Mu Chu</strong>, <strong>Chun-Yu Lei</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 4, 1169--1189.</p><p><strong>Abstract:</strong><br/>
In this paper, we study a $(p,q)$-Laplace equation with singular and critical terms and establish the existence of multiple positive solutions via use of variational methods.
</p>projecteuclid.org/euclid.rmjm/1567044034_20190828220100Wed, 28 Aug 2019 22:01 EDTSolvability of the mixed problem of a high-order PDE with fractional time derivatives, Sturm-Liouville operators on spatial variables and non-local boundary conditionshttps://projecteuclid.org/euclid.rmjm/1567044035<strong>Onur Alp Ilhan</strong>, <strong>Shakirbay G. Kasimov</strong>, <strong>Umrbek S. Madraximov</strong>, <strong>Haci M. Baskonus</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 4, 1191--1206.</p><p><strong>Abstract:</strong><br/>
In this paper, we study the solvability of a mixed problem of a partial differential equation of high order with fractional derivatives with respect to time and with Sturm-Liouville operators with spatial variables and non-local boundary conditions; the solution is found as a series of eigenfunctions of the Sturm-Liouville operator with non-local boundary conditions.
</p>projecteuclid.org/euclid.rmjm/1567044035_20190828220100Wed, 28 Aug 2019 22:01 EDTOn $k$-restricted overpartitionshttps://projecteuclid.org/euclid.rmjm/1567044036<strong>Uha Isnaini</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 4, 1207--1221.</p><p><strong>Abstract:</strong><br/>
We introduce $k$-restricted overpartitions, which are generalizations of overpartitions. In such partitions, among those parts of the same magnitude, one of the first $k$ occurrences may be overlined. We first give the generating function and establish the $5$-dissections of $k$-restricted overpartitions. Then we provide a combinatorial interpretation for certain Ramanujan type congruences modulo $5$. Finally, we pose some problems for future work.
</p>projecteuclid.org/euclid.rmjm/1567044036_20190828220100Wed, 28 Aug 2019 22:01 EDTThe ascending chain condition on principal ideals in composite generalized power series ringshttps://projecteuclid.org/euclid.rmjm/1567044037<strong>Jung Wook Lim</strong>, <strong>Dong Yeol Oh</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 4, 1223--1236.</p><p><strong>Abstract:</strong><br/>
Let $D \subseteq E$ be an extension of commutative rings with identity, $I$ a nonzero proper ideal of $D$, $(\Gamma , \leq )$ a strictly totally ordered monoid such that $0 \leq \alpha $ for all $\alpha \in \Gamma $, and $\Gamma ^*=\Gamma \setminus \{0\}$. Let $D+[\![E^{\Gamma ^*, \leq }]\!]=\{f \in [\![E^{\Gamma , \leq }]\!] \mid f(0) \in D\}$ and $D+[\![I^{\Gamma ^*, \leq }]\!] =\{f \in [\![D^{\Gamma , \leq }]\!] \mid f(\alpha ) \in I$ for all $\alpha \in \Gamma ^*\}$. In this paper, we give some conditions for the rings $D+[\![E^{\Gamma ^*, \leq }]\!]$ and $D+[\![I^{\Gamma ^*, \leq }]\!]$ to satisfy the ascending chain condition on principal ideals.
</p>projecteuclid.org/euclid.rmjm/1567044037_20190828220100Wed, 28 Aug 2019 22:01 EDTExistence and uniqueness of solution for a nonhomogeneous discrete fractional initial value problemhttps://projecteuclid.org/euclid.rmjm/1567044038<strong>A. Khastan</strong>, <strong>H. Azadi</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 4, 1237--1257.</p><p><strong>Abstract:</strong><br/>
This paper is devoted to the study of a nonhomogeneous discrete fractional initial value problem. Using the Laplace transform, we present the existence and uniqueness of the solution. We illustrate the applicability of results by an example.
</p>projecteuclid.org/euclid.rmjm/1567044038_20190828220100Wed, 28 Aug 2019 22:01 EDTSpectral triples for nonarchimedean local fieldshttps://projecteuclid.org/euclid.rmjm/1567044039<strong>Slawomir Klimek</strong>, <strong>Sumedha Rathnayake</strong>, <strong>Kaoru Sakai</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 4, 1259--1291.</p><p><strong>Abstract:</strong><br/>
Using associated trees, we construct a spectral triple for the $\mathrm {C}^*$-algebra of continuous functions on the ring of integers $R$ of a nonarchimedean local field $F$ of characteristic zero, and investigate its properties. Remarkably, the spectrum of the spectral triple operator is closely related to the roots of a $q$-hypergeometric function. We also study a noncompact version of this construction for the $\mathrm {C}^*$-algebra of continuous functions on $F$, vanishing at infinity.
</p>projecteuclid.org/euclid.rmjm/1567044039_20190828220100Wed, 28 Aug 2019 22:01 EDTThe integer group determinants for the symmetric group of degree fourhttps://projecteuclid.org/euclid.rmjm/1567044040<strong>Christopher Pinner</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 4, 1293--1305.</p><p><strong>Abstract:</strong><br/>
For the symmetric group $S_4$ we determine all the integer values taken by its group determinant when the matrix entries are integers.
</p>projecteuclid.org/euclid.rmjm/1567044040_20190828220100Wed, 28 Aug 2019 22:01 EDTWell-posedness of semilinear strongly damped wave equations with fractional diffusion operators and $C^0$ potentials on arbitrary bounded domainshttps://projecteuclid.org/euclid.rmjm/1567044041<strong>Joseph L. Shomberg</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 4, 1307--1334.</p><p><strong>Abstract:</strong><br/>
We examine the well-posedness of a strongly damped wave equation equipped with fractional diffusion operators. Ranges on the orders of the diffusion operators are determined in connection with global well-posedness of mild solutions or the global existence of weak solutions. Local existence proofs employ either semigroup methods or a Faedo-Galerkin scheme, depending on the type of solution sought. Mild solutions arising from semigroup methods are either analytic or of Gevrey class; the former produce a gradient system. We also determine the critical exponent for the nonlinear term depending on the orders of the fractional diffusion operators. Thanks to the nonlocal presentation of the fractional diffusion operators, we are able to work on arbitrary bounded domains. The nonlinear potential is only assumed to be continuous while satisfying a suitable growth condition.
</p>projecteuclid.org/euclid.rmjm/1567044041_20190828220100Wed, 28 Aug 2019 22:01 EDTConvexity with respect to weakly $q$-concave line bundles and reductionhttps://projecteuclid.org/euclid.rmjm/1567044042<strong>Viorel Vâjâitu</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 4, 1335--1354.</p><p><strong>Abstract:</strong><br/>
We reconsider intermediate holomorphic convexity introduced by Barlet and Silva in the more general case of convexity with respect to hermitian holomorphic line bundles whose weight functions are (weakly) $q$-convex. Also we give completeness and Kahler properties of the canonical reduction.
</p>projecteuclid.org/euclid.rmjm/1567044042_20190828220100Wed, 28 Aug 2019 22:01 EDTTheory and analysis of partial differential equations with a $\psi $-Caputo fractional derivativehttps://projecteuclid.org/euclid.rmjm/1567044043<strong>D. Vivek</strong>, <strong>E.M. Elsayed</strong>, <strong>K. Kanagarajan</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 4, 1355--1370.</p><p><strong>Abstract:</strong><br/>
In this paper, we study existence and stability of solutions to a partial differential equation with $\psi $-fractional derivative. Existence results are established by means of the fixed point and upper and lower solution methods. In addition, Ulam-type stability results are derived by using the Gronwall inequality method. Finally, an example is provided to illustrate the theoretical results.
</p>projecteuclid.org/euclid.rmjm/1567044043_20190828220100Wed, 28 Aug 2019 22:01 EDTInfinitely many sign-changing solutions for the Hardy-Sobolev-Maz'ya equation involving critical growthhttps://projecteuclid.org/euclid.rmjm/1567044044<strong>Lixia Wang</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 4, 1371--1390.</p><p><strong>Abstract:</strong><br/>
In this paper, we consider the existence of infinitely many sign-changing solutions for the following Hardy-Sobolev-Maz'ya equation \[ \left \{\begin{aligned} &-\triangle u-\frac {\mu u}{|y|^2}=\lambda u+\frac {|u|^{2^{\ast }(t)-2}u}{|y|^t} \quad \mbox {in } \Omega , \\ &u=0 \quad \qquad \qquad \qquad \qquad \qquad \qquad \,\mbox {on } \partial \Omega , \end{aligned} \right . \] where $\Omega $ is an open bounded domain in $\mathbb {R}^N, \mathbb {R}^N=\mathbb {R}^k\times \mathbb {R}^{N-k}$, $\lambda >0$, $0\leq \mu \lt {(k-2)^2}/{4}$ when $k>2$, $\mu =0$ when $k=2$ and $2^{\ast }(t)={2(N-t)}/({N-2})$. A point $x\in \mathbb {R}^N$ is denoted as $x=(y,z)\in \mathbb {R}^k\times \mathbb {R}^{N-k}$, and the points $x^0=(0,z^0)$ are contained in $\Omega $. By using a compactness result obtained previously, we prove the existence of infinitely many sign changing solutions by a combination of the invariant sets method and the Ljusternik-Schnirelman type minimax method.
</p>projecteuclid.org/euclid.rmjm/1567044044_20190828220100Wed, 28 Aug 2019 22:01 EDTProjective dimension and regularity of edge ideals of some weighted oriented graphshttps://projecteuclid.org/euclid.rmjm/1567044045<strong>Guangjun Zhu</strong>, <strong>Li Xu</strong>, <strong>Hong Wang</strong>, <strong>Zhongming Tang</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 4, 1391--1406.</p><p><strong>Abstract:</strong><br/>
We provide formulas for the projective dimension and the regularity of edge ideals associated to vertex weighted rooted forests and oriented cycles. We derive from that exact formulas for the depth of those ideals. We also give some examples to show that the assumptions cannot be dropped.
</p>projecteuclid.org/euclid.rmjm/1567044045_20190828220100Wed, 28 Aug 2019 22:01 EDTPerturbed obstacle problems in Lipschitz domains: linear stability and nondegeneracy in measurehttps://projecteuclid.org/euclid.rmjm/1568880084<strong>Ivan Blank</strong>, <strong>Jeremy LeCrone</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 5, 1407--1418.</p><p><strong>Abstract:</strong><br/>
We consider the classical obstacle problem on bounded, connected Lipschitz domains $D \subset \mathbb R^n$. We derive quantitative bounds on the changes to contact sets under general perturbations to both the right-hand side and the boundary data for obstacle problems. In particular, we show that the Lebesgue measure of the symmetric difference between two contact sets is linearly comparable to the $L^1$-norm of perturbations in the data.
</p>projecteuclid.org/euclid.rmjm/1568880084_20190919040153Thu, 19 Sep 2019 04:01 EDTSymmetric diophantine systems and families of elliptic curves of high rankhttps://projecteuclid.org/euclid.rmjm/1568880087<strong>Ajai Choudhry</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 5, 1419--1447.</p><p><strong>Abstract:</strong><br/>
While there has been considerable interest in the problem of finding elliptic curves of high rank over $\mathbb {Q}$, very few parametrized families of elliptic curves of generic rank $\geq 8$ have been published. In this paper we use solutions of certain symmetric diophantine systems to construct a number of families of elliptic curves whose coefficients are given in terms of several arbitrary parameters and whose generic rank ranges from at least 8 to at least 12. Specific numerical values of the parameters yield elliptic curves with quite large coefficients and we could therefore determine the precise rank only in a few cases where the rank of the elliptic curve $\leq 13$. It is, however, expected that the parametrized families of elliptic curves obtained in this paper would yield examples of elliptic curves of much higher rank.
</p>projecteuclid.org/euclid.rmjm/1568880087_20190919040153Thu, 19 Sep 2019 04:01 EDTLattice-ordered groups generated by an ordered group and regular systems of idealshttps://projecteuclid.org/euclid.rmjm/1568880088<strong>Thierry Coquand</strong>, <strong>Henri Lombardi</strong>, <strong>Stefan Neuwirth</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 5, 1449--1489.</p><p><strong>Abstract:</strong><br/>
Unbounded entailment relations, introduced by Paul Lorenzen (1951), are a slight variant of a notion which plays a fundamental role in logic (Scott 1974) and in algebra (Lombardi and Quitte 2015). We call systems of ideals their single-conclusion counterpart. If they preserve the order of a commutative ordered monoid $G$ and are equivariant with respect to its law, we call them equivariant systems of ideals for $G$ : they describe all morphisms from $G$ to meet-semilattice-ordered monoids generated by (the image of) $G$. Taking a 1953 article by Lorenzen as a starting point, we also describe all morphisms from a commutative ordered group $G$ to lattice-ordered groups generated by $G$ through unbounded entailment relations that preserve its order, are equivariant, and satisfy a regularity property invented by Lorenzen; we call them regular entailment relations . In particular, the free lattice-ordered group generated by $G$ is described through the finest regular entailment relation for $G$, and we provide an explicit description for it; it is order-reflecting if and only if the morphism is injective, so that the Lorenzen-Clifford-Dieudonné theorem fits into our framework. Lorenzen's research in algebra starts as an inquiry into the system of Dedekind ideals for the divisibility group of an integral domain $R$, and specifically into Wolfgang Krull's ``Fundamentalsatz'' that $R$ may be represented as an intersection of valuation rings if and only if $R$ is integrally closed: his constructive substitute for this representation is the regularisation of the system of Dedekind ideals, i.e. the lattice-ordered group generated by it when one proceeds as if its elements are comparable.
</p>projecteuclid.org/euclid.rmjm/1568880088_20190919040153Thu, 19 Sep 2019 04:01 EDTOn the projective dimension of $5$ quadric almost complete intersections with low multiplicitieshttps://projecteuclid.org/euclid.rmjm/1568880089<strong>Sabine El Khoury</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 5, 1491--1546.</p><p><strong>Abstract:</strong><br/>
Let $S$ be a polynomial ring over an algebraically closed field $k$ and $ \mathfrak p =(x,y,z,w) $ a homogeneous height $4$ prime ideal. We give a finite characterization of the degree $2$ component of ideals primary to $\mathfrak p$, with multiplicity $e \leq 3$. We use this result to give a tight bound on the projective dimension of almost complete intersections generated by five quadrics with $e \leq 3$.
</p>projecteuclid.org/euclid.rmjm/1568880089_20190919040153Thu, 19 Sep 2019 04:01 EDTOn the integer transfinite diameter of intervals of the form $ \left [ \frac {r}{s}, u \right ] $ or $[0, (\sqrt {a}- \sqrt {b})^2 ]$ and of Farey intervalshttps://projecteuclid.org/euclid.rmjm/1568880090<strong>V. Flammang</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 5, 1547--1562.</p><p><strong>Abstract:</strong><br/>
We consider intervals of the form $\left [ \frac {r}{s}, u \right ] $, where $r,s$ are positive integers with gcd($r,s$)=1 and $u$ is a real number, or of the form $[0, \smash {(\sqrt {a}{-}\sqrt {b})^2} ]$, where $a,b$ are positive integers. Thanks to a lemma of Chudnovsky, we give first a lower bound of the integer transfinite diameter of such intervals. Then, using the method of explicit auxiliary functions and our recursive algorithm, we explain how to get an upper bound for this quantity. We finish with some numerical examples. Secondly, we prove inequalities on the integer transfinite diameter of Farey intervals, i.e., intervals of the type $\bigl [ \frac {a}{q}, \frac {b}{s} \bigr ] $, where $|as-bq|=1$.
</p>projecteuclid.org/euclid.rmjm/1568880090_20190919040153Thu, 19 Sep 2019 04:01 EDTA uniqueness theorem for the Nica--Toeplitz algebra of a compactly aligned product systemhttps://projecteuclid.org/euclid.rmjm/1568880091<strong>James Fletcher</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 5, 1563--1594.</p><p><strong>Abstract:</strong><br/>
Fowler introduced the notion of a product system: a collection of Hilbert bimodules $\mathbf {X}=\{\mathbf {X}_p:p\in P\}$ indexed by a semigroup $P$, endowed with a multiplication implementing isomorphisms $\mathbf {X}_p\otimes _A \mathbf {X}_q\cong \mathbf {X}_{pq}$. When $P$ is quasi-lattice ordered, Fowler showed how to associate a $C^*$-algebra $\mathcal {NT}_\mathbf {X}$ to $\mathbf {X}$, generated by a universal representation satisfying some covariance condition. In this article we prove a uniqueness theorem for these so called Nica–Toeplitz algebras.
</p>projecteuclid.org/euclid.rmjm/1568880091_20190919040153Thu, 19 Sep 2019 04:01 EDTExistence and smoothness results for a new class of $n$-dimensional Navier-Stokes equationshttps://projecteuclid.org/euclid.rmjm/1568880092<strong>Rim Jday</strong>, <strong>Zennir Khaled</strong>, <strong>Svetlin G. Georgiev</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 5, 1595--1615.</p><p><strong>Abstract:</strong><br/>
We propose and develop a new approach to prove the global existence of solutions for a class of $n$-dimensional Navier-Stokes equations.
</p>projecteuclid.org/euclid.rmjm/1568880092_20190919040153Thu, 19 Sep 2019 04:01 EDTExistence of traveling wave solutions in a stage structured cooperative system on higher-dimensional latticeshttps://projecteuclid.org/euclid.rmjm/1568880093<strong>Kun Li</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 5, 1617--1631.</p><p><strong>Abstract:</strong><br/>
We study the existence of traveling wave solutions in a higher-dimensional lattice cooperative system with stage structure. We establish the existence theorem of traveling wave solutions based on the upper and lower solutions method and Schauder's fixed point theorem. Then we construct a pair of upper and lower solutions to verify the existence of traveling wave solutions.
</p>projecteuclid.org/euclid.rmjm/1568880093_20190919040153Thu, 19 Sep 2019 04:01 EDTExponents of primitive companion matriceshttps://projecteuclid.org/euclid.rmjm/1568880094<strong>Monimala Nej</strong>, <strong>A. Satyanarayana Reddy</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 5, 1633--1645.</p><p><strong>Abstract:</strong><br/>
A nonnegative matrix $A$ is primitive if for some positive integer $m$ all entries in $A^m$ are positive. The smallest such $m$ is called the exponent of $A$ and written $\exp (A)$. For the class of primitive companion matrices $X$, we find $\exp (A)$ for certain $A \in X$. We find certain values of $m$ for which there is an $n \times n$ primitive companion matrix (for given $n$) with exponent $m$. We also propose open problems for further research.
</p>projecteuclid.org/euclid.rmjm/1568880094_20190919040153Thu, 19 Sep 2019 04:01 EDTSeparability properties of singular degenerate abstract differential operators and applicationshttps://projecteuclid.org/euclid.rmjm/1568880095<strong>Veli B. Shakhmurov</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 5, 1647--1666.</p><p><strong>Abstract:</strong><br/>
We study separability and spectral properties of singular degenerate elliptic equations in vector-valued $L_{p}$ spaces. We prove that a realization operator according to this equation with some boundary conditions is separable and Fredholm in $L_{p}$. The leading part of the associated differential operator is not self-adjoint. The sharp estimate of the resolvent, discreteness of spectrum and completeness of root elements of this operator is obtained. Moreover, we show that this operator is positive and generates a holomorphic $C_{0}$-semigroups on $L_{p}$. In application, we examine the regularity properties of nonlocal boundary value problem for degenerate elliptic equation and for the system of degenerate elliptic equations of either finite or infinite number.
</p>projecteuclid.org/euclid.rmjm/1568880095_20190919040153Thu, 19 Sep 2019 04:01 EDTMeans, moments and Newton's inequalitieshttps://projecteuclid.org/euclid.rmjm/1568880096<strong>R. Sharma</strong>, <strong>A. Sharma</strong>, <strong>R. Saini</strong>, <strong>G. Kapoor</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 5, 1667--1677.</p><p><strong>Abstract:</strong><br/>
It is shown that Newton's inequalities and the related Maclaurin's inequalities provide several refinements of the fundamental arithmetic-geometric-harmonic mean inequality and Sierpinski's inequality in terms of the means and variance of positive real numbers. We also obtain some inequalities involving third and fourth central moments of real numbers.
</p>projecteuclid.org/euclid.rmjm/1568880096_20190919040153Thu, 19 Sep 2019 04:01 EDTEffective divisor classes of a projective plane bundle over an elliptic curvehttps://projecteuclid.org/euclid.rmjm/1568880097<strong>Tomokuni Takahashi</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 5, 1679--1708.</p><p><strong>Abstract:</strong><br/>
We consider generators of a monoid consisting of effective divisor classes in Neron-Severi group for each isomorphism class of a projective plane bundle over an elliptic curve. Furthermore, we investigate the structure of the effective divisors which are algebraically equivalent to integral multiples of the generators of the monoid.
</p>projecteuclid.org/euclid.rmjm/1568880097_20190919040153Thu, 19 Sep 2019 04:01 EDTGlobal existence and exponential decay for a viscoelastic Petrovsky systemhttps://projecteuclid.org/euclid.rmjm/1568880098<strong>Yaojun Ye</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 5, 1709--1724.</p><p><strong>Abstract:</strong><br/>
We investigate the initial-boundary value problem for a viscoelastic Petrovsky system of Kirchhoff-type with nonlinear source term. By applying the standard continuation principle, the existence of global solutions for this problem is proved and the exponential decay estimate of global solutions is obtained.
</p>projecteuclid.org/euclid.rmjm/1568880098_20190919040153Thu, 19 Sep 2019 04:01 EDTExistence of solutions for quasilinear Kirchhoff type problems with critical nonlinearity in $\mathbb {R}^N$https://projecteuclid.org/euclid.rmjm/1568880099<strong>Jing Zhang</strong>, <strong>Alatancang Chen</strong>. <p><strong>Source: </strong>Rocky Mountain Journal of Mathematics, Volume 49, Number 5, 1725--1753.</p><p><strong>Abstract:</strong><br/>
We study the existence of solutions for a class of Kirchhoff type problems with critical growth in $\mathbb {R}^N$:$$-\varepsilon ^2\biggl (a+b\int _{\mathbb {R}^N}|\nabla u|^2\,dx\biggr )\Delta u + V(x)u -\varepsilon ^2a\Delta (u^2)u = |u|^{22^\ast -2}u + h(x,u),$$ $(t, x) \in \mathbb {R} \times \mathbb {R}^N$. By using a change of variables, the quasilinear equations are reduced to a semilinear one, whose associated functionals are well defined in the usual Sobolev space and satisfy the geometric conditions of the mountain pass theorem for suitable assumptions. We prove that it has at least one solution and for any $m \in \mathbb {N}$, it has at least $m$ pairs of solutions. The proofs are based on the variational methods and concentration-compactness principle.
</p>projecteuclid.org/euclid.rmjm/1568880099_20190919040153Thu, 19 Sep 2019 04:01 EDT